It adds to India's north-south divide. (I think John Dalman's answer--Hindu nationalism--is the main reason why the possible name change is controversial. Here I just want to add another important reason why the name change might be controversial.)
In Hindi, India is simply Bharat. So, Bharat is already commonly used in north Indians to refer to India.
In contrast, while south Indians do use Bharat (or variants) to refer to India, they do so less often. Tamil speakers might instead use இந்தியா (Intiya) and Malayalam speakers might instead use ഇന്ത്യ (Inthya).
According to one writer Devdutta Pattanaik (India Today, Business Today):
The word ‘Bharat’ refers to certain parts of North India. It’s a name given by Brahmins ... So, whenever people say Bharat, Bharatavarsha, Bharatakhanda, these are coming from Brahmin sources, like Vedas, Puranas, Dharmashastras. It is a very North Indian, Brahminical word that emerges.
Now Rig Veda happens only in the Kurukshetra region. You have the Mahabharata epic, again talking about the Bharata clan…and it happens only in the northern part of India. Kuru-Panchala region is what is today roughly Delhi, Mathura, up to Prayagraj. First, the word Bharatvarsh was carved in stone about 2100 years ago in Odisha in the Hathigumpha caves. You find this word Bharatvarsh but it is referring only to the Gangetic region, not to the entire India. So, the word Bharatvarsh refers to a certain part of North India…and it is a name given by Brahmins. So, when people say Bharat, Bharatkhand, Bharatvarsh, these are coming from Brahmin sources…something like an Aryadesh, the land of Aryans. Bharata is a victorious king of the Aryans. So, it is a very North Indian Brahminical word that emerges
The News Minute:
Social scientist R. Raghuram while speaking to IANS said: “The name change from India to Bharat will be difficult to accept in south India and the DMK will capitalize on it. As a political front who fans Tamil feelings, this issue will be used across the state for tremendous political gain. The BJP, if it intends to change the name, will have to face the backlash in Tamil Nadu and it is certain that the alliance with the AIADMK will not be of any help as the name Bharat is difficult to resonate here.”
The DMK, with its Dravidian identity of social equality and promotion of Tamil, has seized upon the name change issue and will actively speak against it in the state. There is a possibility of a move like the anti-Hindi agitation being revived up by the party in the state.